The Existence of Positive Solutions to Kirchhoff Type Equations in $\mathbb{R}^{N}$ With Asymptotic Nonlinearity

Hongyu Ye, Fengli Yin

Abstract

In this paper, we are concerned with the following Kirchhoff problem$$\left\{%\begin{array}{ll}\vspace{0.2cm} \left(a+\lambda\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)|u|^2)\right)[-\Delta u+V(x)u]=f(x,u), & \hbox{$x\in \mathbb{R}^N$},\\ u\in H^1(\mathbb{R}^N),~~~~u>0, & \hbox{$x\in \mathbb{R}^N$},\end{array}%\right.$$where $N\geq3$, $a>0$ is a constant, $\lambda>0$ is a parameter, the potential $V(x)$ may not be radially symmetric and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some assumptions on $V$ and $f$, we prove the existence of a positive solution for $\lambda$ small and the nonexistence result for $\lambda$ large.

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